Both versions of OP's example are fine. It's not really relevant that Mom gave both brothers a ride. You can still use either in, say, "My brother stayed home sick, but my/our Mom gave me a ride to school". Provided another sibling has been mentioned in the discourse, our is always okay (but not required).
It's worth pointing out that the word Mom itself clearly indicates an informal context. UK speakers (I'm not sure about US) often use our in informal contexts even if no other sibling has been explicitly mentioned. This is particularly noticeable in the North of England, where you'll hear people use our kid [our Jimmy] to identify their brother [Jimmy]. But such usages would not be considered "formally correct".
Also note that as J.R. says, it's perfectly okay (mainly in informal usages) not to include any possessive pronoun at all with Mom, Mother, Dad, Father, Pop, etc. It's obvious whose parent you're talking about.
It's part of the available subtlety of English that people can choose whether to use the "inclusive" plural possessive, or plain singular. Sometimes you can make an educated guess as to how well a married couple are getting on, for example, by noting how often one of them chooses to use my when referring to something applicable to both of them (my house, my son, etc.).
But of course, some speakers habitually use my in contexts where others might use our, simply because they happen to be more self-centered, or they picked up the habit from a "less-than-inclusive" childhood.
